Bol loops and Bruck loops of order \(pq\) (Q502733)

The paper provides a classification of right Bol loops and right Bruck loops of order \(pq\) up to isomorphism where \(p\) and \(q\) are odd primes such that \(p>q\). Any right Bol loop satisfies the identity \(((zx)y)x=z((xy)x\), and a right Bruck loop is a right Bol loop satisfying the identity \((xy)^{-1}=x^{-1}y^{-1}\). Let \(p\) and \(q\) be odd primes such that \(p>q\), and let \(Q\) denote a right Bol loop of order \(pq\). It is shown by loop-theoretic arguments that \(Q\) contains a unique subloop of order \(p\) which is normal in \(Q\). The authors derive a multiplication formula for \(Q\). The multiplication in \(Q\) is uniquely determined by certain complete mappings of \(\mathbb{Z}_p\). First, the special case of right Bruck loops of order \(pq\) is settled by scrutinizing the groups \(A= \mathrm{Mlt}_r(Q)\rtimes \langle J\rangle \) where \(\mathrm{Mlt}_r(Q)\) is the right multiplication group of the Bruck loop \(Q\) with inversion map \(J\). A nonassociative right Bruck loop of order \(pq\) exists if and only if \(q\) is a divisor of \(p^2-1\), and in such case the loop is unique up to isomorphism. Thus, every right Bruck loop \(Q\) of order \(pq\) is either isomorphic to the cyclic group \(\mathbb{Z}_{pq}\), or \(q\) divides \(p^2-1\) and \(Q\) is isomorphic to the unique nonassociative Bruck loop \(B_{p,q}\) of order \(pq\). The right multiplication group of a nonassociative Bruck loop of order \(pq\) is isomorphic to the semidirect product of \(\mathbb{Z}_p\times \mathbb{Z}_p\) with \(\mathbb{Z}_q\). These results for right Bruck loops of order \(pq\) are generalized to right Bol loops of order \(pq\): When \(Q\) is a nonassociative right Bol loop of order \(pq\) then \(q\) divides \(p^2-1\). The right and the middle nuclei of such loops \(Q\) are trivial, whereas the left nucleus of \(Q\) is normal and isomorphic to \(\mathbb{Z}_p\). The right multiplication group of a right Bol loop of order \(pq\) has order \(p^2q\) or \(p^3q\). Moreover, the complete mappings occurring in the multiplication formula for \(Q\) are linear. Finally, the authors present an abstract construction of all nonassociative right Bol loops of order \(pq\) where \(q\) divides \(p^2-1\). A crucial step in this direction is the solution of the eigenvalue problem for a certain circulant matrix. There are exactly \((p-q+4)/2\) nonassociative right Bol loops of order \(pq\).